<pre class=metadata>
Title: CSS Easing Functions Module Level 2
Status: ED
Prepare for TR: no
Work Status: exploring
Shortname: css-easing
Level: 2
Group: csswg
ED: https://drafts.csswg.org/css-easing/
TR: https://www.w3.org/TR/css-easing-2/
Editor: Brian Birtles, Mozilla https://www.mozilla.org/, bbirtles@mozilla.com, w3cid 43194
Editor: Dean Jackson, Apple Inc https://www.apple.com/, dino@apple.com, w3cid 42080
Editor: Tab Atkins Jr., Google, http://xanthir.com/contact, w3cid 42199
Editor: Chris Lilley, W3C, https://svgees.us/, w3cid 1438
Former Editor: Matt Rakow, Microsoft, w3cid 62267
Former Editor: Shane Stephens, Google, shans@google.com, w3cid 47691
Former Editor: Jake Archibald, Google, jakearchibald@google.com, w3cid 76394
Implementation Report: https://wpt.fyi/results/css/css-easing
Markup Shorthands: markdown yes
Indent: 2

Abstract: This CSS module describes a way for authors to define a transformation
    that controls the rate of change of some value.
    Applied to animations, such transformations can be used to produce
    animations that mimic physical phenomena such as momentum or to cause the
    animation to move in discrete steps producing robot-like movement.
    Level 2 adds more sophisticated functions for custom easing curves.

!Participate: IRC: <a href="ircs://irc.w3.org:6667/css">#css</a> on W3C's IRC
Repository: w3c/csswg-drafts
WPT Path Prefix: css/css-easing/
WPT Display: closed
!Tests: <a href="https://github.com/web-platform-tests/wpt/tree/master/css/css-easing">web-platform-tests css/css-easing</a>
</pre>

<style>
.easing-graph {
  max-width: 400px;
  width: 100%;
}
@media (prefers-color-scheme: dark) {
  .easing-graph {
    background: none;
  }
}
</style>
<style>
/* Creates an easing example. Use like <div class=easing-track style="--easing: linear"></div>. */
.easing-track {
  position: relative;
  height: 1em;
  background: linear-gradient(to bottom, transparent calc(50% - 1px), var(--a-normal-underline) 0 calc(50% + 1px), transparent 0);
  min-width: 200px;
  margin: .5em;
}
.easing-track::before {
  content: "";
  position: absolute;
  width: 1em;
  height: 1em;
  box-sizing: border-box;
  border-radius: 50%;
  background: var(--bg);
  border: medium solid var(--text);
  left: 0;
  animation: --easing-track var(--easing) 3s infinite;
}
@keyframes --easing-track {
  to { left: calc(100% - 1em); }
}
</style>
<style>
/* Put nice boxes around each algorithm. */
[data-algorithm]:not(.heading) {
  padding: .5em;
  border: thin solid #ddd; border-radius: .5em;
  margin: .5em calc(-0.5em - 1px);
}
[data-algorithm]:not(.heading) > :first-child {
  margin-top: 0;
}
[data-algorithm]:not(.heading) > :last-child {
  margin-bottom: 0;
}
[data-algorithm] [data-algorithm] {
    margin: 1em 0;
}
</style>

Introduction {#introduction}
============================

<em>This section is not normative.</em>

It is often desirable to control the rate at which some value changes.
For example, gradually increasing the speed at which an element moves can
give the element a sense of weight as it appears to gather momentum.
This can be used to produce intuitive user interface elements or convincing
cartoon props that behave like their physical counterparts.
Alternatively, it is sometimes desirable for animation to move forwards in
distinct steps such as a segmented wheel that rotates such that the segments
always appear in the same position.

Similarly, controlling the rate of change of gradient interpolation can be
used to produce different visual effects such as suggesting a concave or convex
surface, or producing a striped effect.

[=Easing functions=] provide a means to transform such values by taking an
input progress value and producing a corresponding transformed output progress
value.

<figure>
  <img class="easing-graph" src="images/easing-function-example.svg" width="535" height="510"
    alt="Example of an easing function that produces an ease-in effect.">
  <figcaption>
    Example of an easing function that produces an ease-in effect.<br>
    Given an input progress of 0.7, the easing function scales the
    value to produce an output progress of 0.52.<br>
    Applying this easing function to an animation would cause it to progress
    more slowly at first but then gradually progress more quickly.
  </figcaption>
</figure>

<h3 id="values">
Value Definitions</h3>

  This specification uses the <a href="https://www.w3.org/TR/css-values-3/#value-defs">value definition syntax</a> from [[!CSS-VALUES-3]].
  Value types not defined in this specification are defined in CSS Values &amp; Units [[!CSS-VALUES-3]].
  Combination with other CSS modules may expand the definitions of these value types.

<h2 id=easing-functions oldids=timing-functions>Easing functions</h2>

An <dfn export lt="easing function|timing function">easing function</dfn> takes
an [=input progress value=] and produces an [=output progress value=].

An [=easing function=] must be a pure function meaning that for a given set of
inputs, it always produces the same [=output progress value=].

The <dfn export for="easing function">input progress value</dfn> is a real number in the range [-&infin;,
&infin;].
Typically, the [=input progress value=] is in the range [0, 1] but this may
not be the case when [=easing functions=] are chained together.

<div class=note>

An example of when easing functions are chained together occurs in
Web Animations [[WEB-ANIMATIONS]]
where the output of the easing function specified on an animation effect
may become the input to an easing function specified on
one of the keyframes of a keyframe effect.
In this scenario,
the input to the easing function on the keyframe effect
may be outside the range [0, 1].

</div>

The <dfn export for="easing function">output progress value</dfn> is a real number in the
range [-&infin;, &infin;].

Note: While CSS numbers have a theoretically infinite range
(see [[css-values-4#numeric-ranges]])
UAs will automatically clamp enormous numbers to a reasonable range.
If easing functions are used outside of the CSS context,
care must be taken to either correctly handle potential infinities
(including those produced by merely <em>very large</em> values
stored in a floating point number),
or clamp the [=output progress value=].

Some types of easing functions
also take an additional boolean <dfn export for="easing function">before flag</dfn>,
which indicates the easing has not yet started,
or is going in reverse and is past the finish.
(Some easing functions can have multiple possible [=output progress values=]
for a given [=input progress value=],
and generally favor the last one specified;
this flag instead causes those [=easing functions=]
to favor the first specified value
before the animation has started.)

This specification defines several types of easing functions:

<pre class="prod">
  <dfn>&lt;easing-function&gt;</dfn> = <<linear-easing-function>>
                    | <<cubic-bezier-easing-function>>
                    | <<step-easing-function>>
</pre>

<wpt>
  timing-functions-syntax-computed.html
  timing-functions-syntax-invalid.html
  timing-functions-syntax-valid.html
</wpt>

<!-- Big Text: linear 

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█████ ████ █▌   ▐▌ █████▌ █▌  █▌ █▌  █▌
-->

<h3 id=the-linear-easing-function>
Linear Easing Functions: ''linear'', ''linear()''</h3>

A <dfn export>linear easing function</dfn>
is an [=easing function=]
that interpolates linearly
between its [=linear()/control points=].
Each <dfn for="linear()">control point</dfn>
is a pair of numbers,
associating an [=input progress value=]
to an [=output progress value=].

<figure>
  <img class="easing-graph" src="images/linear-easing-curve.svg" width="535" height="510"
      alt="A linear curve used as an easing function.">
  <figcaption>
    ''linear(0, .1 25%, .75 50%, 1)''<br>
    The shape of the curve follows the [=linear()/control points=].<br>
    Input progress values serve as <var ignore>x</var> values of the curve,
    whilst the <var ignore>y</var> values are the output progress values.
  </figcaption>
</figure>

A [=linear easing function=] has the following syntax:

<pre class="prod">
  <dfn>&lt;linear-easing-function></dfn> = linear | <<linear()>>
  <dfn>linear()</dfn> = linear( [ <<number>> && <<percentage>>{0,2} ]# )
</pre>

<wpt>
  linear-timing-functions-syntax.html
</wpt>

: ''linear''
:: Equivalent to ''linear(0, 1)''

: ''linear()''
::
  Specifies a [=linear easing function=].

  Each comma-separated argument specifies one or two [=linear()/control points=],
  with an [=input progress value=] equal to the specified <<percentage>>
  (converted to a <<number>> between 0 and 1),
  and an [=output progress value=] equal to the specified <<number>>.
  When the argument has two <<percentage>>s,
  it defines two [=linear()/control points=] in the specified order,
  one per <<percentage>>.

  If an argument lacks a <<percentage>>,
  its [=input progress value=] is initially empty.
  This is corrected at [=used value=] time
  by [=linear() canonicalization=].

<div algorithm>
  To <dfn export lt="linear() canonicalization|canonicalize a linear()">canonicalize a linear()</dfn> function's [=linear()/control points=],
  perform the following:

  1. If the first [=control point=] lacks an [=input progress value=],
    set its [=input progress value=] to 0.

  2. If the last [=control point=] lacks an [=input progress value=],
    set its [=input progress value=] to 1.

  3. If any [=control point=] has an [=input progress value=]
    that is less than the [=input progress value=] of any preceding [=control point=],
    set its [=input progress value=] to the largest [=input progress value=]
    of any preceding [=control point=].

  4. If any [=control point=] still lacks an [=input progress value=],
    then for each contiguous run of such [=linear()/control points=],
    set their [=input progress values=]
    so that they are evenly spaced
    between the preceding and following [=linear()/control points=]
    with [=input progress values=].

  After canonicalization,
  every [=control point=] has an [=input progress value=],
  and the [=input progress values=] are monotonically non-decreasing
  along the list.

  Note: Serialization relies on whether or not an [=input progress value=]
  was originally supplied,
  so that information should be retained
  in the [=CSS/internal representation=].
  It does not rely on whether a pair of [=linear()/control points=]
  were specified as two percentages on a single argument
  or as separate arguments.
</div>

<wpt>
linear-timing-functions-chrome-406926307-crash.html
</wpt>


<h4 id=linear-easing-function-serializing>
Serializing</h4>

The ''linear'' keyword is serialized as itself.

<div algorithm>
  To <dfn export>serialize a linear() function</dfn>:

  1. Let |s| be the string "linear(".
  2. [=serialize a linear() control point|Serialize each control point=] of the function,
    [=string/concatenate=] the results using the separator ", ",
    and append the result to |s|.
  3. Append ")" to |s|,
    and return it.
</div>

<div algorithm>
  To <dfn export>serialize a linear() control point</dfn>:

  1. Let |s| be the serialization,
    as a <<number>>,
    of the [=control point's=] [=output progress value=].

  2. If the [=control point=] originally lacked an [=input progress value=],
    return |s|.

  3. Otherwise, append " " (U+0020 SPACE) to |s|,
    then serialize the [=control point's=] [=input progress value=]
    as a <<percentage>>
    and append it to |s|.

  4. Return |s|.
</div>

<div class=example>
  When serialized,
  [=linear()/control points=] originally specified with two [=input progress values=]
  are turned into two separate [=linear()/control points=],
  and the [=input progress values=] are in strictly ascending order.
  For example:

    - ''linear(0, 0.25, 1)'' serializes as ''linear(0, 0.25, 1)''
    - ''linear(0 20%, 0.5 10%, 1)'' serializes as ''linear(0 20%, 0.5 20%, 1)''
    - ''linear(0, 0.25 25% 75%, 1)'' serializes as ''linear(0, 0.25 25%, 0.25 75%, 1)''
</div>


<h4 id=linear-easing-function-output>
Output</h4>

<div algorithm>

  To <dfn export>calculate linear easing output progress</dfn>
  for a given [=linear easing function=] |func|,
  an [=input progress value=] |inputProgress|,
  and an optional [=before flag=] (defaulting to false),
  perform the following.
  It returns an [=output progress value=].

  1. Let |points| be |func|'s [=linear()/control points=].

  2. If |points| holds only a single item,
    return the [=output progress value=]
    of that item.

  3. If |inputProgress| matches the [=input progress value=]
    of the first point in |points|,
    and the [=before flag=] is true,
    return the first point's [=output progress value=].

  4. If |inputProgress| matches the [=input progress value=]
    of at least one point in |points|,
    return the [=output progress value=] of the last such point.

  5. Otherwise, find two [=linear()/control points=] in |points|,
    |A| and |B|,
    which will be used for interpolation:

    1. If |inputProgress| is smaller
      than any [=input progress value=] in |points|,
      let |A| and |B| be the first two items in |points|.
      If |A| and |B| have the same [=input progress value=],
      return |A|'s [=output progress value=].

    2. Otherwise, if |inputProgress| is larger
      than any [=input progress value=] in |points|,
      let |A| and |B| be the last two items in |points|.
      If |A| and |B| have the same [=input progress value=],
      return |B|'s [=output progress value=].

    3. Otherwise, let |A| be the last [=linear()/control point=]
      whose [=input progress value=] is smaller than |inputProgress|,
      and let |B| be the first [=linear()/control point=]
      whose [=input progress value=] is larger than |inputProgress|.

  6. Linearly interpolate (or extrapolate) |inputProgress|
    along the line defined by |A| and |B|,
    and return the result.

  <wpt>
    linear-timing-functions-output.html
  </wpt>
</div>


<h4 id=linear-easing-function-examples>
Examples</h4>

<div class=example>
  ''linear()'' allows the definition of easing functions that interpolate linearly between a set of points.

  For example, ''linear(0, 0.25, 1)'' produces an easing function
  that moves linearly from 0, to 0.25, then to 1:

  <figure>
    <img class="easing-graph" src="images/simple-linear-example.svg" width="535" height="510"
      alt="linear(0, 0.25, 1) plotted on a graph">
  </figure>

  <figure>
    <table>
      <thead>
        <tr><th>Easing<th>Example
      <tbody>
        <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
        <tr><td>''linear(0, .25, 1)''<td><div class=easing-track style="--easing:linear(0, .25, 1)"></div>
    </table>
    <figcaption>
      An example of the above, with a plain ''linear'' for contrast.
    </figcaption>
  </figure>
</div>

<div class=example>
  By default, values are spread evenly between entries that don't have an explicit "input".
  Input values can be provided using a <<percentage>>.

  For example, ''linear(0, 0.25 75%, 1)'' produces the following easing function,
  which spends 75% of the time transitioning from ''0'' to ''.25'',
  then the last 25% transitioning from ''.25'' to ''1'':

  <figure>
    <img class="easing-graph" src="images/linear-with-input-example.svg" width="535" height="510"
      alt="linear(0, 0.25 75%, 1) plotted on a graph.
        The graph has three points.
        The first is at 0,0.
        The second is at 0.75,0.25.
        The third is at 1,1.">
  </figure>

  <figure>
    <table>
      <thead>
        <tr><th>Easing<th>Example
      <tbody>
        <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
        <tr><td>''linear(0, .25 75%, 1)''<td><div class=easing-track style="--easing:linear(0, .25 75%, 1)"></div>
    </table>
    <figcaption>
      An example of the above, with a plain ''linear'' for contrast.
    </figcaption>
  </figure>
</div>

<div class=example>
  If two input values are provided for a single output,
  it results in two points with the same output,
  causing the easing to "pause" between the two inputs.

  For example, ''linear(0, 0.25 25% 75%, 1)''
  is equivalent to ''linear(0, 0.25 25%, 0.25 75%, 1)'',
  producing the following easing function:

  <figure>
    <img class="easing-graph" src="images/linear-with-double-input-example.svg" width="535" height="510"
      alt="linear(0, 0.25 75%, 1) plotted on a graph.
        The graph has four points.
        The first is at 0,0.
        The second is at 0.25,0.25.
        The third is at 0.75,0.25.
        The forth is at 1,1.">
  </figure>

  <figure>
    <table>
      <thead>
        <tr><th>Easing<th>Example
      <tbody>
        <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
        <tr><td>''linear(0, 0.25 25% 75%, 1)''<td><div class=easing-track style="--easing:linear(0, 0.25 25% 75%, 1)"></div>
    </table>
    <figcaption>
      An example of the above, with a plain ''linear'' for contrast.
    </figcaption>
  </figure>
</div>

<div class=example>
  If the input is outside the range provided,
  the trajectory of the nearest two points is continued.

  For example, here are the implicit values from the previous function:

  <figure>
    <img class="easing-graph" src="images/linear-with-double-input-example-continued.svg" width="535" height="510"
      alt="linear(0, 0.25 75%, 1) plotted on a graph.
        The graph has four points.
        The first is at 0,0.
        The second is at 0.25,0.25.
        The third is at 0.75,0.25.
        The forth is at 1,1.
        The ends of the graph are extended at the angle of the nearest two lines.">
  </figure>
</div>

<div class=example>
  A typical use of ''linear()'' is to provide many points to create the illusion of a curve.

  For example, here's how ''linear()'' could be used to create a reusable "bounce" easing function:

  ```css
  :root {
    --bounce: linear(
      /* Start to 1st bounce */
      0, 0.063, 0.25, 0.563, 1 36.4%,
      /* 1st to 2nd bounce */
      0.812, 0.75, 0.813, 1 72.7%,
      /* 2nd to 3rd bounce */
      0.953, 0.938, 0.953, 1 90.9%,
      /* 3rd bounce to end */
      0.984, 1 100% 100%
    );
  }

  .example {
    animation-timing-function: var(--bounce);
  }
  ```

  The definition ends `1 100% 100%` to create two final points,
  so inputs greater than 1 always output 1.

  <figure>
    <img class="easing-graph" src="images/linear-bounce-example.svg" width="535" height="510"
      alt="The graph of a rough bounce easing.">
  </figure>

  <figure>
    <table>
      <thead>
        <tr><th>Easing<th>Example
      <tbody>
        <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
        <tr><td>''linear(...)''<td><div class=easing-track style="--easing:linear(0, 0.063, 0.25, 0.563, 1 36.4%, 0.812, 0.75, 0.813, 1 72.7%, 0.953, 0.938, 0.953, 1 90.9%, 0.984, 1 100% 100% )"></div>
    </table>
    <figcaption>
      An example of the above, with a plain ''linear'' for contrast.
    </figcaption>
  </figure>

  More points could be used to create a smoother result,
  which may be needed for slower animations.
</div>


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<h3 id=cubic-bezier-easing-functions oldids=cubic-bezier-timing-functions>
Cubic Bézier Easing Functions:
''ease'', ''ease-in'', ''ease-out'', ''ease-in-out'', ''cubic-bezier()''</h3>

A <dfn export>cubic Bézier easing function</dfn>
is an [=easing function=] that interpolates smoothly from 0 to 1
using a cubic polynomial,
influenced by two control points
that the curve will approach
but (usually) not actually reach.
(The "endpoints" of the cubic Bézier
are fixed at (0,0) and (1,1), respectively.)

<figure>
  <img class="easing-graph" src="images/cubic-bezier-easing-curve.svg" width="535" height="510"
      alt="A cubic Bezier curve used as an easing function.">
  <figcaption>
    A cubic Bézier curve used as an easing function.<br>
    The shape of the curve is determined by the location of the control
    points <var ignore>P1</var> and <var ignore>P2</var>.<br>
    Input progress values serve as <var ignore>x</var> values of the curve,
    whilst the <var ignore>y</var> values are the output progress values.
  </figcaption>
</figure>

A <a>cubic Bézier easing function</a> has the following syntax:

<pre class="prod">
  <dfn>&lt;cubic-bezier-easing-function></dfn> =
    <l>''ease''</l> | <l>''ease-in''</l> | <l>''ease-out''</l> | <l>''ease-in-out''</l> | <<cubic-bezier()>>

  <dfn>cubic-bezier()</dfn> = cubic-bezier( [ <<number [0,1]>>, <<number>> ]#{2} )
</pre>

The meaning of each value is as follows:

<dl dfn-type="value" dfn-for="<cubic-bezier-easing-function>">
  : <dfn>ease-in</dfn>
  :: A function that starts slowly and smoothly, then quickly approaches the endpoint with an almost linear curve.
    Equivalent to ''cubic-bezier(0.42, 0, 1, 1)''.
  : <dfn>ease-out</dfn>
  :: A function that starts quickly with an almost linear curve, then slowly and smoothly approaches the endpoint.
    Equivalent to ''cubic-bezier(0, 0, 0.58, 1)''.
  : <dfn>ease-in-out</dfn>
  :: A function that starts and ends slowly and smoothly, quickly traversing the middle part.
    Equivalent to ''cubic-bezier(0.42, 0, 0.58, 1)''.
  : <dfn>ease</dfn>
  :: Similar to ''ease-in-out'', but with a quicker start and a slower finish.
    Equivalent to ''cubic-bezier(0.25, 0.1, 0.25, 1)''.
  : ''cubic-bezier( x1, y1, x2, y2 )''
  ::  Specifies a <a>cubic Bézier easing function</a>.
      The <var ignore>x1</var> and <var ignore>y1</var> arguments
      specify the first control point,
      and <var ignore>x2</var> and <var ignore>y2</var> arguments
      specify the second control point.

      Both <var ignore>x</var> values must be in the range [0, 1]
      or the definition is invalid.

      <details class=note>
        <summary>Details on cubic Bézier curves</summary>

        Note that this does <em>not</em> use the [=input progress value=] as the "t" value
        commonly used to parametrize cubic Bézier curves
        (producing a 2d point as the output),
        but rather uses it as the "x" value on the graph
        (producing a y value as the output).
        This means that only the <em>shape</em> of the curve matters,
        not the velocity along that curve.
        For example, ''cubic-bezier(0, 0, 0, 0)'' and ''cubic-bezier(1, 1, 1, 1)''
        produce exactly the same (linear) easing,
        despite the first's <em>velocity</em> following a <code>t<sup>3</sup></code> curve,
        while the second follows a <code>t<sup>1/3</sup></code> curve.

        <em>In general</em>, cubic Bézier curves can have loops:
        places where a single x value is associated with multiple y values.
        The restriction placed on the control points
        (that their x values be in the [0,1] range)
        prevent this,
        so the resulting easing function is well-defined.
      </details>
</dl>

The keyword values listed above are illustrated below.

<figure>
  <img class="easing-graph" src="images/curve-keywords.svg" width="500"
      alt="The easing functions produced by keyword values.">
  <figcaption>
    The easing functions produced by each of the cubic Bézier easing
    function keyword values.
  </figcaption>
</figure>

<figure>
  <table>
    <thead>
      <tr><th>Easing<th>Example
    <tbody>
      <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
      <tr><td>''ease-in''<td><div class=easing-track style="--easing:ease-in"></div>
      <tr><td>''ease-out''<td><div class=easing-track style="--easing:ease-out"></div>
      <tr><td>''ease-in-out''<td><div class=easing-track style="--easing:ease-in-out"></div>
      <tr><td>''ease''<td><div class=easing-track style="--easing:ease"></div>
  </table>
  <figcaption>
    The above easing functions in action, each applied to a 3s animation of the circles' 'left' property. A ''linear'' easing is provided for contrast.
  </figcaption>
</figure>

<h4 id="bezier-serialization">
Serializing</h4>

The ''ease-in'', ''ease-out'', ''ease-in-out'', and ''ease'' keywords
serialize as themselves.

<div algorithm>
  To <dfn export>serialize a cubic-bezier() function</dfn>:

  1. Let |s| be the string "cubic-bezier(".

  2. Serialize the function's four arguments as <<number>>s,
    [=concatenate=] the results using the separator ", ",
    and append the result to |s|.

  3. Append ")" to |s|, and return it.
</div>


<h4 id=cubic-bezier-algo>
Output</h4>

<div algorithm>
  To <dfn export lt="calculate cubic Bézier easing output progress|calculate cubic Bezier easing output progress">calculate cubic Bézier easing output progress</dfn>
  for a given [=cubic Bézier easing function=] |func|
  and an [=input progress value=] |inputProgress|,
  perform the following.
  It returns an [=output progress value=].

  1. Let |p0| be the point (0,0),
    |p1| be the point given by |func|'s first two arguments,
    |p2| be the point given by |func|'s second two arguments,
    and |p3| be the point (1,1).

  2. If |inputProgress| is within the range [0,1] (inclusive),
    return the y value
    corresponding to |inputProgress| as an x value
    for the cubic Bézier curve
    defined as having |p0| and |p3| as endpoints,
    and |p1| and |p2| as control points.

    The evaluation of this curve is covered in many sources,
    such as [[FUND-COMP-GRAPHICS]].

  3. Otherwise, the curve is extended infinitely,
    using the tangent of the curve at the endpoints.
    This tangent is defined as the line between two points,
    |t1| and |t2|.

    * If |inputProgress| is less than 0,
      let |t1| be |p0|.

      1. If the x value of |p1| is greater than 0,
        let |t2| be |p1|.
      2. Otherwise, if the x value of |p2| is greater than 0,
        let |t2| be |p2|.
      3. Otherwise, return 0.

    * If |inputProgress| is greater than 1,
      let |t2| be |p3|.

      1. If the x value of |p2| is less than 1,
        let |t1| be |p2|.
      2. Otherwise, if the x value of |p1| is less than 1,
        let |t1| be |p1|.
      3. Otherwise, return 1.

    Return the y value
    corresponding to |inputProgress| as an x value
    for the line passing through |t1| and |t2|.

  <wpt>
    cubic-bezier-timing-functions-output.html
  </wpt>
</div>



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<h3 id=step-easing-functions oldids=step-timing-functions>
Step Easing Functions:
''step-start'', ''step-end'', ''steps()''</h3>

A <dfn>step easing function</dfn> is an [=easing function=]
that divides the input time into a specified number of intervals that
are equal in length,
and holds the output steady within each of those intervals.
It is defined by a number of <dfn for=steps()>steps</dfn>, and a <dfn for=steps()>step position</dfn>.
It has the following syntax:

<pre class="prod">
  <dfn>&lt;step-easing-function></dfn> = <l>''step-start''</l> | <l>''step-end''</l> | <<steps()>>

  <dfn>steps()</dfn> = steps( <<integer>>, <<step-position>>?)
  <dfn type>&lt;step-position></dfn> = <l>''jump-start''</l> | <l>''jump-end''</l> | <l>''jump-none''</l> | <l>''jump-both''</l>
                | <l>''start''</l> | <l>''end''</l>
</pre>

<wpt>
  step-timing-functions-syntax.html
</wpt>

The meaning of each value is as follows:

<dl dfn-type=value dfn-for="<step-easing-function>, steps()">
  :   <dfn>step-start</dfn>
  :: Jumps from the starting to the ending value
    at the start of the easing interval.

    Computes to ''steps(1, start)''
  :   <dfn>step-end</dfn>
  :: Jumps from the starting to the ending value
    at the end of the easing interval.

    Computes to ''steps(1, end)''

      <figure>
        <img class="easing-graph" src="images/step-easing-keyword-examples.svg" width="500"
            alt="Example step easing keywords.">
        <figcaption>
          Example step easing function keyword values.
        </figcaption>
      </figure>

  :   ''steps( &lt;integer&gt;, &lt;step-position&gt;? )''
  :: Divides the input interval into a number of equal steps
    specified by the <<integer>>.
    Within each interval, the [=output progress value=] is constant,
    and is determined according to the <<step-position>> keyword.
    If omitted, the <<step-position>> keyword defaults to ''end''.

    If the <<step-position>> is ''jump-none'',
    the <<integer>> must be at least 2,
    or the function is invalid.
    Otherwise,
    the <<integer>> must be at least 1,
    or the function is invalid.

    The <<step-position>> keywords are:

    <dl dfn-type=value dfn-for="<step-position>, steps()">
      : <dfn>jump-start</dfn>
      :: The first interval has an [=output progress value=] of <code>1/steps</code>,
        and subsequent intervals rise by <code>1/steps</code>.

        (It "jumps at the start", with no step returning 0.)
      : <dfn>jump-end</dfn>
      :: The first interval has an [=output progress value=] of <code>0</code>,
        and subsequent intervals rise by <code>1/steps</code>.

        (It "jumps at the end", with no step returning 1.)
      : <dfn>jump-none</dfn>
      :: The first interval has an [=output progress value=] of <code>0</code>,
        and subsequent intervals rise by <code>1/(steps-1)</code>.

        (It "never jumps", with steps returning both 0 and 1.)
      : <dfn>jump-both</dfn>
      :: The first interval has an [=output progress value=] of <code>1/(steps+1)</code>,
        and subsequent intervals rise by <code>1/(steps+1)</code>.

        (It "jumps at both ends", with no steps returning 0 or 1.)
      : <dfn>start</dfn>
      ::  Behaves as ''jump-start''.
      : <dfn>end</dfn>
      ::  Behaves as ''jump-end''.
    </dl>
</dl>

The ''jump-*'' keywords values are illustrated below:

<figure>
  <img class="easing-graph" src="images/step-easing-func-examples.svg" width="500"
      alt="Example step easing functions.">
  <figcaption>
    Example step easing functions.
  </figcaption>
</figure>

<figure>
  <table>
    <thead>
      <tr><th>Easing<th>Example
    <tbody>
      <tr><td>''linear''<td><div class=easing-track style="--easing:linear"></div>
      <tr><td>''steps(3, jump-start)''<td><div class=easing-track style="--easing:steps(3, jump-start)"></div>
      <tr><td>''steps(3, jump-end)''<td><div class=easing-track style="--easing:steps(3, jump-end)"></div>
      <tr><td>''steps(3, jump-none)''<td><div class=easing-track style="--easing:steps(3, jump-none)"></div>
      <tr><td>''steps(3, jump-both)''<td><div class=easing-track style="--easing:steps(3, jump-both)"></div>
  </table>
  <figcaption style="max-width: 80%;">
    The above easing functions in action, with a ''linear'' provided for comparison.
    Each ''steps(3, ...)'' function divides the animation into three constant periods;
    they differ only on what the value is within each period.
  </figcaption>
</figure>




<h4 id=steps-serialization>
Serializing</h4>

Unlike the other [=easing function=] keywords,
''step-start'' and ''step-end'' <em>do not</em> serialize as themselves.
Instead, they serialize as "steps(1, start)" and "steps(1)", respectively.

<div algorithm>
  To <dfn export>serialize a steps() functions</dfn>:

  1. Let |s| be the string "steps(".

  2. Serialize the function's [=steps()/steps=],
    and append it to |s|.

  3. If the function's [=steps()/step position=] is ''end'' or ''jump-end'',
    append ")" to |s| and return |s|.

  4. Otherwise, append ", " to |s|.
    Serialize the [=steps()/step position=] as a keyword,
    and append the result to |s|.
    Append ")" to |s|.
    Return |s|.

  <wpt>
    timing-functions-syntax-computed.html
  </wpt>
</div>


<h4 id=step-easing-algo oldids=step-timing-function-algo>
Output</h4>

<div algorithm>
  To <dfn export>calculate step easing output progress</dfn>
  for a given [=step easing function=] |func|,
  an [=input progress value=] |inputProgress|,
  and an optional [=before flag=] (defaulting to false),
  perform the following.
  It returns an [=output progress value=].

  1. If the [=before flag=] is true, return 0.

  2. Let |steps| be |func|'s [=steps()/steps=],
    and |position| be |func|'s [=steps()/step position=].

  3. Divide the interval [-&infin;, &infin;] into several segments,
    each with an associated value,
    as follows:

    1. [-&infin;, 0) has the value 0.
    2. [1, &infin;] has the value 1.
    3. [0, 1) is divided into |steps| half-open intervals,
      [0, ...) to [..., 1)
      with their values assigned as defined for the |position|
      (see <<step-position>>).

    Note: In other words,
    at the boundary between intervals,
    the associated value is the higher value.

  4. Return the associated value
    for the interval that |inputProgress| is in.

  <wpt>
    step-timing-functions-output.html
  </wpt>
</div>


<h2 class=no-num id=privacy>Privacy Considerations</h2>

<!-- Horizontal review wants Security and Privacy sections to be separate. -->

No new privacy considerations have been reported on this specification.

This specification does not directly introduce any new capabilities to the
Web platform but rather provides common definitions that may be referenced by
other specifications.

<h2 class=no-num id=security>Security Considerations</h2>

Specifications referencing the features defined in this specification
should consider that while easing functions most commonly
take an [=input progress value=] in the range [0,1]
and produce an [=output progress value=] in the range [0, 1],
this is not always the case.
Applications of easing functions should
define the behavior for inputs and outputs outside this range
to ensure they do not introduce new security considerations.

Changes {#changes}
=======

<h3 id="changes-20240828">
Changes since the FPWD of <a href="https://www.w3.org/TR/2024/WD-css-easing-2-20240828/">28 August 2024</a>
</h3>

<ul>
  <li></li>
</ul>

<h3 id="changes-L1">
Additions Since Level 1</h3>

- Added ''linear()'' function.

Acknowledgements {#acknowledgements}
================

This specification is based on the <a
href="https://www.w3.org/TR/css3-transitions/">CSS Transitions</a> specification
edited by L.  David Baron, Dean Jackson, David Hyatt, and Chris Marrin.
The editors would also like to thank Douglas Stockwell, Steve Block, Tab Atkins,
Rachel Nabors, Martin Pitt, and the <a
href="https://damp-lake-50659.herokuapp.com/">Animation at Work</a> slack
community for their feedback and contributions.
